On the nonlinear regularized Schrödinger equation in exterior domains
type: Item
instance of: scholarly article
title: On the nonlinear regularized Schrödinger equation in exterior domains
zbMATH Open document ID: 1471.35258
author: Mohamed Karmous
author: M. Khenissi
published in: Tunisian Journal of Mathematics
publication date: 2021-05-20T00:00:00Z
review text: Let \(\Theta\subset\mathbb R^d\), \(d\in\{2,3\}\), be a compact domain with a smooth boundary. Consider the exterior domain \(\Omega=\mathbb R^d\setminus\Theta\) and the following nonlinear regularized Schrödinger equation \[ (*)\quad\left\{\begin{aligned}i\partial_t u-\Delta_D+i a(x)(-\Delta_D)^{1/2} a(x) u&=F(u)&\quad (t>0,x\in\Omega)\\ u(0,x)&=u_0(x)&\\ u(t,x)&=0&\quad(x\in\Omega)\end{aligned}\right. \] where \(u\) is the unkown function, \(u_0\in C_0^\infty(\Omega), a\in C^\infty(\bar{\Omega})\) are given functions, and \(-\Delta_D\) is the Dirichelt Laplace operator on \(\Omega\). The function \(F\) is non-linear and satisfies \(F(0)=0\). \\ Using the Strichartz estimates from [\textit{M. Karmous}, J. Abstr. Differ. Equ. Appl. 6, No. 1, 37--50 (2015; Zbl 1326.35303)], the authors of the paper under review establish the following two results \begin{itemize} \item[i.] Local well-posedness in \(H^1_0(\Omega)\) for solutions of \((*)\) \item[ii.] Global existence and uniqueness in \(L^2(\Omega)\) for solutions of \((*)\) \end{itemize} For each of the foregoing results i. and ii. to hold, the function \(a\) and the nonlinear interaction \(F\) are supposed to satisfy certain hypotheses.
reviewed by: Ayman Kachmar
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B20
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35Q41
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35A02
zbMATH DE Number: 7350277
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: exterior domain
zbMATH Keywords: exterior geometric control condition
zbMATH Keywords: Kato smoothing effect
zbMATH Keywords: Strichartz estimates
MaRDI profile type: Publication
full work available at URL: tunis.2021.3.361
OpenAlex ID: W3111672045
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DOI: 10.2140/TUNIS.2021.3.361
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description: scientific article; zbMATH DE number 7350277